Exercise 18.4
Page no 18.26
Question 1
Insert five G.M.'s between 16 and $\frac{1}{4}$.
Question 2
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
Question 3
Insert three numbers between 1 and 256 so that the resulting sequence is a G.P.
Question 4
Insert six G.M.'s between $\frac{8}{27}$ and $-5 \frac{1}{16}$.
Question 5
The sum of two numbers is six times their geometric mean. Show that the numbers are in the ratio $3+2 \sqrt{2}: 3-2 \sqrt{2}$.
Question 6
If odd number of G.M.s are inserted between two given quantities $a$ and $b$, show that the middle $\mathrm{G}, \mathrm{M} .=\sqrt{a b}$.
Question 7
(i) If A.M. and GM. of two positive numbers $a$ and $b$ are 10 and 8 respectively, find the numbers.
(ii) If $A$ be the A.M. and $G$ the GM. between two numbers, show that the numbers are $A+\sqrt{A^{2}-G^{2}}$ and $A-\sqrt{A^{2}-G^{2}}$
Question 8
If the ratio of A.M. and G.M. between two numbers $a$ and $b$ is $m: n$, prove that $a: b=m+\sqrt{m^{2}-n^{2}}: m-\sqrt{m^{2}-n^{2}}$.
$[H O T S]$
Question 9
If one G.M., $G$ and two A.M.'s $p$ and $q$ be inserted between two given quantities, prove that $G^{2}=(2 p-q)(2 q-p)$.
|HOTS|
Question 10
If one A.M. A and two GM.'s $p$ and $q$ bc inserted between two given numbers, show that $\frac{p^{2}}{q}+\frac{q^{2}}{p}=2 A$.
$[\mathrm{HOTS}]$
Question 11
If $a$ is the A.M. between $b$ and $c, b$ the G.M. between $a$ and $c$, then show that $\frac{1}{a}, \frac{1}{c}, \frac{1}{b}$ are in A.P.
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